Formal matrix representations of pseudo-Frobenius and Frobenius rings

Abstract

Rings with Nakayama permutations, pseudo-Frobenius and Frobenius rings in particular, are studied by applying the general theory of formal matrix rings to their Peirce decompositions. A combinatorial criterion is given to decide whether a formal matrix ring with local rings on the diagonal has a prescribed Nakayama permutation. It is shown that a pseudo-Frobenius ring R can be represented as a block matrix ring, where the blocks on the diagonal are pseudo-Frobenius rings corresponding to cycles in the Nakayama permutation of R. All possible supports of such blocks are characterised. In the finite case, their local corner rings are shown to be isomorphic. We characterise local corners of quasi-Frobenius rings as a subclass of rings with a Morita self-duality. The duality contexts between these corners then appear on the shifted diagonal of their formal matrix representations. Using the combinatorial criterion, we give, under mild assumptions, a method of how to glue two rings with a Nakayama permutation. It is then used to construct an indecomposable Frobenius ring with two simple modules whose rings of endomorphisms are not isomorphic.